We considered several physically completely different systems, and made sure that the equations of motion are reduced to the same form

Differences between physical systems manifest themselves only in different definitions of the quantity and in a different physical sense of the variable x: it can be a coordinate, an angle, a charge, a current, etc. Note that in this case, as follows from the very structure of equation (1.18), the quantity always has the dimension of inverse time.

Equation (1.18) describes the so-called harmonic vibrations.

The equation of harmonic oscillations (1.18) is a second-order linear differential equation (since it contains the second derivative of the variable x). The linearity of the equation means that

if any function x(t) is a solution to this equation, then the function Cx(t) will also be his solution ( C is an arbitrary constant);

if functions x 1 (t) and x 2 (t) are solutions of this equation, then their sum x 1 (t) + x 2 (t) will also be a solution to the same equation.

A mathematical theorem is also proved, according to which a second-order equation has two independent solutions. All other solutions, according to the properties of linearity, can be obtained as their linear combinations. It is easy to check by direct differentiation that the independent functions and satisfy equation (1.18). So the general solution to this equation is:

where C1,C2 are arbitrary constants. This solution can also be presented in another form. We introduce the quantity

and define the angle as:

Then the general solution (1.19) is written as

According to the trigonometry formulas, the expression in brackets is

We finally arrive at general solution of the equation of harmonic oscillations as:

Non-negative value A called oscillation amplitude, - the initial phase of the oscillation. The whole cosine argument - the combination - is called oscillation phase.

Expressions (1.19) and (1.23) are perfectly equivalent, so we can use either of them for reasons of simplicity. Both solutions are periodic functions of time. Indeed, the sine and cosine are periodic with a period . Therefore, various states of a system that performs harmonic oscillations are repeated after a period of time t*, for which the oscillation phase receives an increment that is a multiple of :

Hence it follows that

The least of these times

called period of oscillation (Fig. 1.8), a - his circular (cyclic) frequency.

Rice. 1.8.

They also use frequency hesitation

Accordingly, the circular frequency is equal to the number of oscillations per seconds.

So, if the system at time t characterized by the value of the variable x(t), then, the same value, the variable will have after a period of time (Fig. 1.9), that is

The same value, of course, will be repeated after a while. 2T, ZT etc.

Rice. 1.9. Oscillation period

The general solution includes two arbitrary constants ( C 1 , C 2 or A, a), the values ​​of which should be determined by two initial conditions. Usually (though not necessarily) their role is played by the initial values ​​of the variable x(0) and its derivative.

Let's take an example. Let the solution (1.19) of the equation of harmonic oscillations describe the motion of a spring pendulum. The values ​​of arbitrary constants depend on the way in which we brought the pendulum out of equilibrium. For example, we pulled the spring to a distance and released the ball without initial velocity. In this case

Substituting t = 0 in (1.19), we find the value of the constant From 2

The solution thus looks like:

The speed of the load is found by differentiation with respect to time

Substituting here t = 0, find the constant From 1:

Finally

Comparing with (1.23), we find that is the oscillation amplitude, and its initial phase is equal to zero: .

We now bring the pendulum out of equilibrium in another way. Let's hit the load, so that it acquires an initial speed , but practically does not move during the impact. We then have other initial conditions:

our solution looks like

The speed of the load will change according to the law:

Let's put it here:

HARMONIC VIBRATION MOTION

§1 Kinematics of harmonic oscillation

Processes that repeat over time are called oscillations.

Depending on the nature of the oscillatory process and the excitation mechanism, there are: mechanical oscillations (oscillations of pendulums, strings, buildings, the earth's surface, etc.); electromagnetic oscillations (oscillations of alternating current, oscillations of vectors and in an electromagnetic wave, etc.); electromechanical vibrations (vibrations of the telephone membrane, loudspeaker diffuser, etc.); vibrations of nuclei and molecules as a result of thermal motion in atoms.

Let's consider the segment [OD] (radius-vector) making rotational motion around the point 0. The length of |OD| = A . Rotation occurs at a constant angular velocity ω 0 . Then the angle φ between the radius vector and the axisxchanges over time according to the law

where φ 0 is the angle between [OD] and the axis X at the timet= 0. Projection of the segment [OD] onto the axis X at the timet= 0

and at an arbitrary point in time

(1)

Thus, the projection of the segment [OD] on the x axis oscillates along the axis X, and these fluctuations are described by the cosine law (formula (1)).

Oscillations that are described by the cosine law

or sinus

called harmonic.

Harmonic vibrations are periodical, because the value of x (and y) is repeated at regular intervals.

If the segment [OD] is in the lowest position in the figure, i.e. dot D coincides with the point R, then its projection on the x-axis is zero. Let's call this position of the segment [OD] the position of equilibrium. Then we can say that the value X describes the displacement of an oscillating point from its equilibrium position. The maximum displacement from the equilibrium position is called amplitude fluctuations

Value

which stands under the cosine sign is called the phase. Phase determines the displacement from the equilibrium position at an arbitrary point in timet. Phase at the initial moment of timet = 0 equal to φ 0 is called the initial phase.

T

The period of time during which one complete oscillation takes place is called the period of oscillation. T. The number of oscillations per unit time is called the oscillation frequency ν.

After a period of time equal to the period T, i.e. as the cosine argument increases by ω 0 T, the movement is repeated, and the cosine takes the same value

because cosine period is equal to 2π, then, therefore, ω 0 T= 2π

thus, ω 0 is the number of oscillations of the body in 2π seconds. ω 0 - cyclic or circular frequency.

harmonic wave pattern

BUT- amplitude, T- period, X- offset,t- time.

We find the speed of the oscillating point by differentiating the displacement equation X(t) by time

those. speed vout of phase with offset X on theπ /2.

Acceleration - first derivative of velocity (second derivative of displacement) with respect to time

those. acceleration a differs from the phase shift by π.

Let's build a graph X( t) , y( t) and a( t) in one estimate of coordinates (for simplicity, we take φ 0 = 0 and ω 0 = 1)

Free or own oscillations that occur in a system left to itself after it has been taken out of equilibrium are called.

Harmonic oscillation is a phenomenon of periodic change of some quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:

where x is the value of the changing quantity, t is time, the remaining parameters are constant: A is the amplitude of the oscillations, ω is the cyclic frequency of the oscillations, is the full phase of the oscillations, is the initial phase of the oscillations.

Generalized harmonic oscillation in differential form

(Any non-trivial solution of this differential equation is a harmonic oscillation with a cyclic frequency)

Types of vibrations

Free oscillations are performed under the action of the internal forces of the system after the system has been taken out of equilibrium. For free oscillations to be harmonic, it is necessary that the oscillatory system be linear (described by linear equations of motion), and there should be no energy dissipation in it (the latter would cause damping).

Forced oscillations are performed under the influence of an external periodic force. For them to be harmonic, it is sufficient that the oscillatory system be linear (described by linear equations of motion), and the external force itself changes over time as a harmonic oscillation (that is, that the time dependence of this force is sinusoidal).

Harmonic vibration equation

Equation (1)

gives the dependence of the fluctuating value S on time t; this is the equation of free harmonic oscillations in explicit form. However, the equation of oscillations is usually understood as a different record of this equation, in differential form. For definiteness, we take equation (1) in the form

Differentiate it twice with respect to time:

It can be seen that the following relation holds:

which is called the equation of free harmonic oscillations (in differential form). Equation (1) is a solution to differential equation (2). Since equation (2) is a second-order differential equation, two initial conditions are necessary to obtain a complete solution (that is, to determine the constants A and   included in equation (1); for example, the position and speed of an oscillatory system at t = 0.

A mathematical pendulum is an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small eigenoscillations of a mathematical pendulum of length l, motionlessly suspended in a uniform gravitational field with free fall acceleration g, is equal to

and does not depend on the amplitude and mass of the pendulum.

A physical pendulum is an oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.

Along with the translational and rotational motions of bodies in mechanics, oscillatory motions are also of considerable interest. Mechanical vibrations called the movements of bodies that repeat exactly (or approximately) at regular intervals. The law of motion of an oscillating body is given by some periodic function of time x = f (t). The graphic representation of this function gives a visual representation of the course of the oscillatory process in time.

Examples of simple oscillatory systems are a load on a spring or a mathematical pendulum (Fig. 2.1.1).

Mechanical oscillations, like oscillatory processes of any other physical nature, can be free and forced. Free vibrations are made under the influence internal forces system after the system has been brought out of equilibrium. The oscillations of a weight on a spring or the oscillations of a pendulum are free oscillations. vibrations under the action external periodically changing forces are called forced .

The simplest type of oscillatory process are simple harmonic vibrations , which are described by the equation

x = x m cos (ω t + φ 0).

Here x- displacement of the body from the equilibrium position, x m - oscillation amplitude, i.e. the maximum displacement from the equilibrium position, ω - cyclic or circular frequency hesitation, t- time. The value under the cosine sign φ = ω t+ φ 0 is called phase harmonic process. At t= 0 φ = φ 0 , so φ 0 is called initial phase. The minimum time interval after which the movement of the body is repeated is called period of oscillation T. The physical quantity reciprocal to the period of oscillation is called oscillation frequency:

Oscillation frequency f shows how many vibrations are made in 1 s. Frequency unit - hertz(Hz). Oscillation frequency f is related to the cyclic frequency ω and the oscillation period T ratios:

On fig. 2.1.2 shows the positions of the body at regular intervals with harmonic vibrations. Such a picture can be obtained experimentally by illuminating an oscillating body with short periodic flashes of light ( stroboscopic lighting). The arrows represent the velocity vectors of the body at different points in time.

Rice. 2.1.3 illustrates the changes that occur on the graph of a harmonic process if either the amplitude of the oscillations changes x m , or period T(or frequency f), or the initial phase φ 0 .

When the body oscillates along a straight line (axis OX) the velocity vector is always directed along this straight line. Velocity υ = υ x body movement is determined by the expression

In mathematics, the procedure for finding the limit of the ratio at Δ t→ 0 is called the calculation of the derivative of the function x (t) by time t and denoted as or as x"(t) or finally as . For the harmonic law of motion Calculation of the derivative leads to the following result:

The appearance of the term + π / 2 in the cosine argument means a change in the initial phase. Maximum modulo values ​​of velocity υ = ω x m are achieved at those moments of time when the body passes through the equilibrium positions ( x= 0). Acceleration is defined in a similar way a = ax bodies with harmonic vibrations:

hence the acceleration a is equal to the derivative of the function υ ( t) by time t, or the second derivative of the function x (t). The calculations give:

The minus sign in this expression means that the acceleration a (t) always has the opposite sign of the offset x (t), and, therefore, according to Newton's second law, the force that causes the body to perform harmonic oscillations is always directed towards the equilibrium position ( x = 0).

Changes in time according to a sinusoidal law:

where X- the value of the fluctuating quantity at the moment of time t, BUT- amplitude , ω - circular frequency, φ is the initial phase of oscillations, ( φt + φ ) is the total phase of oscillations . At the same time, the values BUT, ω and φ - permanent.

For mechanical vibrations with an oscillating value X are, in particular, displacement and speed, for electrical oscillations - voltage and current strength.

Harmonic oscillations occupy a special place among all types of oscillations, since this is the only type of oscillation whose shape is not distorted when passing through any homogeneous medium, i.e., waves propagating from a source of harmonic oscillations will also be harmonic. Any non-harmonic vibration can be represented as a sum (integral) of various harmonic vibrations (in the form of a spectrum of harmonic vibrations).

Energy transformations during harmonic vibrations.

In the process of oscillations, there is a transition of potential energy Wp into kinetic W k and vice versa. In the position of maximum deviation from the equilibrium position, the potential energy is maximum, the kinetic energy is zero. As we return to the equilibrium position, the speed of the oscillating body increases, and with it the kinetic energy also increases, reaching a maximum in the equilibrium position. The potential energy then drops to zero. Further-neck movement occurs with a decrease in speed, which drops to zero when the deflection reaches its second maximum. Potential energy here increases to its initial (maximum) value (in the absence of friction). Thus, the oscillations of the kinetic and potential energies occur with a doubled (compared to the oscillations of the pendulum itself) frequency and are in antiphase (i.e., there is a phase shift between them equal to π ). Total vibration energy W remains unchanged. For a body oscillating under the action of an elastic force, it is equal to:

where v m- the maximum speed of the body (in the equilibrium position), x m = BUT- amplitude.

Due to the presence of friction and resistance of the medium, free oscillations damp out: their energy and amplitude decrease with time. Therefore, in practice, not free, but forced oscillations are used more often.